Matrices

Let A= bmatrix1 & 0 & 0\ 3 & 1 & 0\ 9 & 3 & 1 bmatrix and B= [bij ], 1 i, j 3. If B=A99-I, then the value of b31-b21b32 is :

1. 99
2. 199
3. 149
4. 159

Let M be a 3 3 matrix such that M bmatrix1\ 0\ 0 bmatrix= bmatrix1\ 2\ 3 bmatrix, M bmatrix0\ 1\ 0 bmatrix= bmatrix0\ 1\ 2 bmatrix and M bmatrix0\ 0\ 1 bmatrix= bmatrix-1\ 1\ 1 bmatrix. If M bmatrixx\ y\ z bmatrix= bmatrix1\ 7\ 11 bmatrix, then x+y+z equals :

1. 4
2. 5
3. 7
4. 11

Let S= \A= bmatrixa & b\ c & d bmatrix : a, b, c, d \0, 1, 2, 3, 4\ and A2-4 A+3I=0 \ be a set of 2 2 matrices. Then the number of matrices in S, for which the sum of the diagonal elements is equal to 4, is :

1. 20
2. 17
3. 21
4. 19

Let A= bmatrix1 & 2\ 1 & bmatrix and B= bmatrix3 & 3\ & 2 bmatrix. If A2-4A+I=0 and B2-5B-6I=0, then among the two statements: (S1) : [(B-A)(B+A) ]T= bmatrix13 & 15\ 7 & 10 bmatrix and (S2) : (adj(A+B))=-5

1. only (S1) is correct
2. only (S2) is correct
3. both (S1) and (S2) are correct
4. both (S1) and (S2) are wrong

Consider the matrices A= bmatrix2 & -2\ 4 & -2 bmatrix and B= bmatrix3 & 9\ 1 & 3 bmatrix. If matrices P and Q are such that PA=B and AQ=B, then the absolute value of the sum of the diagonal elements of 2(P+Q) is .

Let A, B and C be three 2 × 2 matrices with real entries such that B = (I + A)-1 and A + C = I. If BC = bmatrix 1 & -5 \ -1 & 2 bmatrix and CB bmatrix x1 \ x2 bmatrix = bmatrix 12 \ -6 bmatrix, then x1 + x2 is

1. 4
2. 2
3. 0
4. −2

Let A = bmatrix 3 & -4 \ 1 & -1 bmatrix and B be two matrices such that A100 = 100B + I. Then the sum of all the elements of B100 is

The number of 3 2 matrices A, which can be formed using the elements of the set \-2, -1, 0, 1, 2\ such that the sum of all the diagonal elements of ATA is 5, is

Let A = bmatrix 0 & 2 & -3 \ -2 & 0 & 1 \ 3 & -1 & 0 bmatrix and B be a matrix such that B(I - A) = I + A. Then the sum of the diagonal elements of BTB is equal to

If A = bmatrix 2 & 3 \ 3 & 5 bmatrix, then the determinant of the matrix (A2025 - 3 A2024 + A2023 ) is

1. 12
2. 24
3. 28
4. 16

For some , R, let A = bmatrix & 2 \ 1 & 2 bmatrix and B = bmatrix 1 & 1 \ 1 & bmatrix be such that A2 - 4A + 2I = B2 - 3B + I = O. Then ( ( adj(A3 - B3)))2 is equal to .

For the matrices A = bmatrix 3 & -4 \ 1 & -1 bmatrix and B = bmatrix -29 & 49 \ -13 & 18 bmatrix, if (A15 + B) bmatrix x \ y bmatrix = bmatrix 0 \ 0 bmatrix, then among the following which one is true?

1. x = 16, y = 3
2. x = 5, y = 7
3. x = 11, y = 2
4. x = 18, y = 11

Let A = bmatrix & 0 & - \ 0 & 1 & 0 \ & 0 & bmatrix. If for some (0, ), A2 = AT, then the sum of the diagonal elements of the matrix (A + I)3 + (A - I)3 - 6A is equal to .

Let the matrix A = bmatrix 1 & 0 & 0 \ 1 & 0 & 1 \ 0 & 1 & 0 bmatrix satisfy An = An-2 + A2 - I for n 3. Then the sum of all the elements of A50 is :

1. 44
2. 39
3. 52
4. 53

Let A = bmatrix & -1 \ 6 & bmatrix, > 0, such that (A) = 0 and + = 1. If I denotes 2 2 identity matrix, then the matrix (I + A)8 is :

1. bmatrix 257 & -64 \ 514 & -127 bmatrix
2. bmatrix 766 & -255 \ 1530 & -509 bmatrix
3. bmatrix 1025 & -511 \ 2024 & -1024 bmatrix
4. bmatrix 4 & -1 \ 6 & -1 bmatrix

Let A be a 3 3 real matrix such that A2(A - 2I) - 4(A - I) = O, where I and O are the identity and null matrices, respectively. If A5 = A2 + A + I, where , , and are real constants, then + + is equal to :

1. 76
2. 12
3. 4
4. 20

Let S = \m Z : Am2 + Am = 3I - A-6 \, where A = bmatrix 2 & -1 \ 1 & 0 bmatrix. Then n(S) is equal to .

Let A = [aij] be a matrix of order 3 3, with aij = ( 2)i+j. If the sum of all the elements in the third row of A2 is + 2, , Z, then + is equal to :

1. 210
2. 280
3. 224
4. 168

Let M denote the set of all real matrices of order 3 3 and let S = \-3, -2, -1, 1, 2\. Let S1 = \A = [aij] M : A = AT and aij S, i, j\, S2 = \A = [aij] M : A = -AT and aij S, i, j\, S3 = \A = [aij] M : a11 + a22 + a33 = 0 and aij S, i, j\. If n(S1 S2 S3) = 125 , then equals .

Let A = bmatrix 1 2 & -2 \ 0 & 1 bmatrix and P = bmatrix & - \ & bmatrix, > 0. If B = PAP , C = P B10P and the sum of the diagonal elements of C is mn, where (m, n) = 1, then m + n is :

1. 127
2. 2049
3. 258
4. 65

Let A be a 3 3 matrix such that XTAX=O for all nonzero 3 1 matrices X= bmatrixx\ \ bmatrix. If A bmatrix1\\1\\1 bmatrix= bmatrix1\\4\\-5 bmatrix, A bmatrix1\\2\\1 bmatrix= bmatrix0\\4\\-8 bmatrix, and (adj(2(A+I)))=2 3 5 , , , N, then 2+ 2+ 2 is

Let A=[aij] be a square matrix of order 2 with entries either 0 or 1. Let E be the event that A is an invertible matrix. Then the probability P(E) is :

1. 38
2. 18
3. 316
4. 58

Let A=[aij] be a 3 3 matrix such that A bmatrix0\\1\\0 bmatrix= bmatrix0\\0\\1 bmatrix, A bmatrix4\\1\\3 bmatrix= bmatrix0\\1\\0 bmatrix and A bmatrix2\\1\\2 bmatrix= bmatrix1\\0\\0 bmatrix, then a23 equals :

1. 2
2. -1
3. 1
4. 0